Let $A,B$ and $C$ be three points on a circle of radius $1$. (a) Show that the area of the triangle $ABC$ equals \[\frac12(\sin(2\angle ABC)+\sin(2\angle BCA)+\sin(2\angle CAB))\] (b) Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle $ABC$ is maximized when $\angle BCA=\angle CAB$ (c) Hence or otherwise, show that the area of the triangle $ABC$ is maximum when the triangle is equilateral.

The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.

Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$

A circle passes through the three vertices of an isosceles triangle that has two sides of length $ 3$ and a base of length $ 2$. What is the area of this circle? $ \textbf{(A)}\ 2\pi\qquad \textbf{(B)}\ \frac {5}{2}\pi\qquad \textbf{(C)}\ \frac {81}{32}\pi\qquad \textbf{(D)}\ 3\pi\qquad \textbf{(E)}\ \frac {7}{2}\pi$

Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$

Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

Given right triangle $ ABC$, with $ AB \equal{} 4, BC \equal{} 3,$ and $ CA \equal{} 5$. Circle $ \omega$ passes through $ A$ and is tangent to $ BC$ at $ C$. What is the radius of $ \omega$?

Real numbers $x$ and $y$ satisfy the system of equalities $$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$ Prove that $\cos 2x = \cos 2y$.

$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.

Find $\lim_{n\to\infty} \int_0^{\pi} e^{x}|\sin nx|dx.$

Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying: \[ \sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k). \]

In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$. [asy] import geometry; import graph; unitsize(1.5 cm); pair A, B, C, D; B = (0,0); D = (3,0); A = 2*dir(120); C = extension(B,dir(30),A,D); draw(A--B--D--cycle); draw(B--C); draw(arc(B,0.5,0,30)); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, NE); label("$D$", D, SE); label("$30^\circ$", (0.8,0.2)); label("$90^\circ$", (0.1,0.5)); perpendicular(B,NE,C-B); [/asy]

In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle

If \[x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2},\] where $ i^2 \equal{} \minus{} 1$, then which of the following is [i]not[/i] correct? $ \textbf{(A)}\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad \textbf{(B)}\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad \textbf{(C)}\ x^9 \plus{} y^9 \equal{} \minus{} 1$ $ \textbf{(D)}\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad \textbf{(E)}\ x^{13} \plus{} y^{13} \equal{} \minus{} 1$

Evaluate \[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]

Let $x$ be the least real number greater than $1$ such that $\sin(x)$ = $\sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer? $\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20$

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

Let triangle $ ABC$ acutangle, with $ m \angle ACB\leq\ m \angle ABC$. $ M$ the midpoint of side $ BC$ and $ P$ a point over the side $ MC$. Let $ C_{1}$ the circunference with center $ C$. Let $ C_{2}$ the circunference with center $ B$. $ P$ is a point of $ C_{1}$ and $ C_{2}$. Let $ X$ a point on the opposite semiplane than $ B$ respecting with the straight line $ AP$; Let $ Y$ the intersection of side $ XB$ with $ C_{2}$ and $ Z$ the intersection of side $ XC$ with $ C_{1}$. Let $ m\angle PAX \equal{} \alpha$ and $ m\angle ABC \equal{} \beta$. Find the geometric place of $ X$ if it satisfies the following conditions: $ (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP}$ $ (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}$

For how many angles $x$, in radians, satisfying $0\le x<2\pi$ do we have $\sin(14x)=\cos(68x)$? \[\rm a. ~128\qquad \mathrm b. ~130\qquad \mathrm c. ~132 \qquad\mathrm d. ~134\qquad\mathrm e. ~136\]

Let $M$ be an interior point of the triangle $ABC$ such that $AMC = 90^\circ$, $AMB = 150^\circ$, and $BMC = 120^\circ$. The circumcenters of the triangles $AMC$, $AMB$, and $BMC$ are $P$, $Q$, and $R$ respectively. Prove that the area of $\Delta PQR$ is greater than or equal to the area of $\Delta ABC$.

What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\] $\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84$

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]